Pull the chair legs slightly further apart, and remove the loose rung from the joints in the chair legs. Wrap a piece of carpet around the chair leg for protection, and tap the leg gently in that spot, using a rubber mallet, to pry open stubborn legs that you can't open further by hand.
Sand the ends of the rung with 100-grit sandpaper. Be careful not to reduce the size of the rung ends, but only remove the glue from the ends.
Wrap the sandpaper around the end of the rung. Slide the end of the rung into the leg hole, and twist the rung from side to side, to scrape out the glue inside the hole.
Cut a slot into each end of the rung, using a backsaw. Slide a small, thin wood wedge halfway into each slot. Trim the wedges down to the proper length and width, if necessary, using a utility knife.
Apply carpenter's glue to the ends of the rung, and also into the chair leg holes. Install the rung ends into the chair leg holes.
Wrap the chair leg with a piece of carpet, and gently tap the leg with a rubber mallet, so that the rung seats fully in the holes. This also drives the wedges deeper into the rung slots, expanding the rung ends and creating a tighter fit within the holes. This creates a tight joint and eliminates the wiggle.